Is it permissible to leave one's current job due to racism and gender mixing, knowing that there is a family in need of money, and with the intention that Allah is the Provider and a dislike of vice?

It is permissible to leave work that involves unlawful mixing, and indeed, one should do so. Supporting a wife and children does not make it unlawful to leave such work. However, if no other work or sufficient money for expenses is available, then one must remain in it until another option becomes available, while striving to avoid temptations. And whoever intends to please Allah by leaving the work or remaining in it out of necessity for spending is sincere.